From WhatsApp go to the contact you want to transfer their chat to Telegram, and choose export chat.
There is an easier way where you can export chats from the main interface, drag the chat you want to export to the left, as you can see in the image below click on more options.
Now many options will appear to you, click on export chat, as you can see in the image down below.
After you click export chat, all the options for sharing the chat will normally appear, one of those options is Telegram, click on the icon of the application to continue the process.
On telegram after you export the chats, it will be received as a compressed file, the compressed file contains your chats in WhatsApp with the media and stickers, as you can see in the image below.
This procedure makes it easy for you to transfer your WhatsApp chats to any other application, to continue chatting with your friends without any problem.